Utility grids, such as electrical power grids, might have one or more plants that produce said utility for distribution by the grids. Said plants, however, may be subject to random adverse events that affect their ability to produce a demanded amount of said utility. Some utility grids have reservoirs for storing said utility for distribution to the plants to make up for the adverse events. These distributions of a utility from a reservoir to a plant due to random adverse events are referred to herein as “contingency discharges”. The contingency discharges from a reservoir to one of the plants might occur during a first time period. In order to make up for the contingency discharges during a first time period, the plants connected to the reservoir by the grid may be required to make compensatory charges to the reservoir during a subsequent second time period. A technical problem to be solved, therefore, is controlling the compensatory charges from the plants to the reservoir during the second time period so that in aggregate they make up for the contingency discharges of the first time period.
It is not enough, however, to say that the individual plants should make compensatory charges to the reservoir equal to the contingency discharges they each individually received from the reservoir during the first time period. The contingency discharges may be rare and when they do occur, they may be quite large. This is referred to herein as a “long tail distribution” of contingency discharges. Thus, it might exceed a plant's ability to provide compensatory charges during a second time period when a rare large contingency discharge occurs during a first time period. Thus, a further technical problem to be solved is how to control the compensatory charges from all of the plants connected to the reservoir by the grid so that in aggregate they make up for a distribution of contingency discharges that has large rare events (i.e. has a long tail distribution).